Primary Model Elements
The model contains five primary elements: ideas, performance landscapes, actors, a social network, and change events.
Ideas. Ideas are comprised of multiple features that jointly determine the value of a given idea to the organization. We capture these properties in our model by representing ideas on a N-dimensional idea space, in which each dimension corresponds to a particular feature of an idea, and each feature can have one of two states. Ideas are represented by a N-digit bit string where each digit denotes the state of its corresponding feature, on (1) or off (0).
We also created an option to run the model with a custom, multilevel representation for ideas that enabled us to visualize the model in low dimensions without losing the complexity we desired in the problem space. At the high level of the custom option, ideas are defined by their location in a two-dimensional feature space. Each dimension contains 32, instead of 2, states which are denoted by discrete integer values. At a low level, each combination of features also corresponds to a 10-digit bit string representation of the integer location, which can be thought of as containing more detailed information on the elements of each idea. The bit string is a concatenation of the binary representations of the integers. For example, the 5-digit binary representation of the integer 17 is 10001, and for the integer 25 it is 11001. The 10-digit bit string representation for the location {17, 25} would therefore be 1000111001. The low-level bit string representation is used to recombine ideas in the same way as the non-custom representation of ideas; the high-level integer representation of an idea’s location the feature space is used to determine its performance through a “custom landscape” described below.Performance Landscapes. The performance landscape maps the feature space location of an idea to the performance scores for that idea. A key characteristic of the types of performance landscapes we use in this model is that the impact of a particular value of one dimension of the feature space on performance depends on the value of the other. The more complex the problem (i.e., the more interdependencies matter for performance), the greater the number of peaks and valleys in the landscape; therefore, the easier it is to get stuck at local optima and not find the highest performing location (Levinthal, 1997). Each peak in the landscape has a unique basin of attraction, which is defined as the set of all locations that would lead to that peak if actors could engage only in self-search that always chose the adjacent location with the maximum performance. The basin of attraction for the peak with the highest performance we consider “the last mile.”
Performance landscapes were generated using two broad approaches. The first follows the common “NK landscape” approach used in the computational strategy and organizations literature, where N represents the number of dimensions in idea space and K the degree of interdependence between those dimensions (Levinthal, 1997; Lazar & Friedman, 2007; Lenox, Rockart, & Lewin, 2007). Each feature, f, makes a contribution to performance, cf. cf itself is comprised of multiple components. The number of components in cf is determined by the number of other features on which f depends, K, and the number of states possible for each feature – one component is included for every possible combination of states among the interconnected features. cf is calculated by summing the performance value of each of its constituent components. The performance value of an entire solution is determined by averaging across all cf. In our implementation, the K other features to which f was connected were selected at random, and the performance value of each component within cf was determined by a random draw from a uniform distribution ranging from 1 to 1000. A new NK landscape was generated for each run of the model. An overview of their mean characteristics is provided in Table OA1.
The second approach was used only with our “custom” idea representation. When the custom option was used, we generated a new performance landscape for each run by picking five percent of the locations as starting points for diffusing performance to surrounding areas, and one additional location as a clear global peak. Figure 1a in the main text depicts a representative custom landscape. On average, the custom landscapes used in the model contained 9.6 total peaks (ranging from 5 to 15), and the highest peak of a landscape featured the level of idea performance that was 3.5 times as high as that of the second highest peak.
Table OA1. Characteristics of Performance Landscapes Used in Model | |||||
|---|---|---|---|---|---|
Mean value from 5000 runs of the model | |||||
Landscape | Number of solutions | Local peaks (number) | Local peaks (% of solutions) | Size of the last mile (% of solutions) | Reaching Global Peak (% of focal action chains) |
Custom Landscape | 1,024 | 9.61 | 0.94 | 10.82 | 17.06 |
NK Landscape, N=8, K=2 | 256 | 5.21 | 2.03 | 48.07 | 36.56 |
NK Landscape, N=8, K=4 | 256 | 10.43 | 4.07 | 27.67 | 22.18 |
NK Landscape, N=8, K=6 | 256 | 19.70 | 7.70 | 13.99 | 12.48 |
NK Landscape, N=10, K=2 | 1,024 | 7.65 | 0.75 | 44.08 | 32.50 |
NK Landscape, N=10, K=4 | 1,024 | 18.32 | 1.79 | 23.13 | 19.24 |
NK Landscape, N=10, K=6 | 1,024 | 37.67 | 3.68 | 11.19 | 11.02 |
NK Landscape, N=12, K=2 | 4,096 | 11.89 | 0.29 | 39.76 | 29.34 |
NK Landscape, N=12, K=4 | 4,096 | 32.44 | 0.79 | 19.50 | 17.72 |
NK Landscape, N=12, K=6 | 4,096 | 75.95 | 1.85 | 8.65 | 9.50 |
- Actors. Actors hold a belief about what location in the
idea space yields the highest performance. An actor can update their
current belief with a higher performing belief in one of three ways.
First, they can search the areas of the idea space that are adjacent to
their current belief on their own (self-search). For NK landscapes,
adjacent is defined by all the solutions that result from switching the
state of a single feature from 0 to 1, or vice versa. For the custom
landscapes, adjacent is defined as one of a solutions eight neighbors on
the solution grid. The actors self-search behavior is governed by a
parameter called ss-max that equaled true when actors picked
the maximum performing adjacent solution when engaged in self-search,
and false when actors randomly picked any better performing adjacent
solution. In all cases, the solution picked had to be higher performing
than their existing solution. Second, they can copy a higher performing
belief from a network contact (transfer). Third, they can recombine
elements of their current belief with those of a network contact.
Actors are also characterized by two general tendencies that drive the actions happening in a model iteration (see Figure OA1). One tendency, collab, is an actor’s tendency to collaborate with others. It governs the frequency with which the actor attempts to engage in collaboration (i.e., recombination or copying) as opposed to searching on their own. The second tendency, recomb, is the tendency to innovate when working with others. It governs the frequency with which the actor attempts to engage in recombination instead of copying, given that an actor has engaged in collaboration. In the results presented in the main text, the values of collab and recomb for a specific actor were determined by taking a random draw from a normal distribution with a mean 0.8 and standard deviation of 0.1. This implies that the expected probability that self-search occurs in any given round is 0.2, and the expected probability of having two actors willing to collaborate is (0.8)×(0.8) = 0.64. By the same token, since recomb is set to 0.8, it implies that the expected probability that two actors engage in a recombination interaction in any given round is P(interaction)×recomb×recomb = (0.64)×(0.8)×(0.8) = 0.4096. Similarly, the expected probability that a transfer occurs in any given round is (0.64)×(0.2) = 0.128. Note that engaging in a search or interaction activity of any type does not mean that either actor changed his or her belief. Beliefs are updated only if a better-performing idea is developed or found. Our boundary condition analyses, reported in the main text, explore the sensitivity of our results to varying the mean levels of collab and recomb.
Figure OA1. Description of a Model Iteration
Notes: Ego is the actor randomly selected to act in a given iteration,
while alter corresponds to any connected actor that ego chooses to
collaborate with. Paremeters collabi and collabj
represent ego’s and alter’s propensities to collaborate, respectively,
while recombi and recombj represent ego’s and
alter’s propensities to recombine, respectively.
- Social network. A social network of collegial ties governs
the interactions between actors and the possible transfers and
recombinations. We used one of five different network conditions to
generate a network at initialization. Three of at the network conditions
generated small-world networks with 45 total edges as described in the
main text, wherein 25 actors were distributed into 5 tightly
interconnected network communities that were only sparsely
interconnected to each other (SW1, SW2, SW3, Figure 2); the “random”
network condition generated networks that randomly assigned the same
number of edges between workers; and the “fully connected” condition
where all workers were connected to all other workers. Conditions SW2,
SW3, and “random” involved random rewiring of edges, resulting in the
generation of a range of networks with similar properties. Conditions
SW1 and “fully connected” involved no rewiring so they resulted in
exactly the same network structure every time.
The social network also governed the initialization of beliefs. Network communities were randomly assigned a different point in the feature space. Actors within a network community were then randomly assigned a location on or near that point. Consequently, actors hold ideas much more similar to others within their network community than to those outside their network community when the simulation begins. The only restriction on the point that served as the centroid for the network community was it could not be a location from which an actor could find the global peak through only self-search. (Relaxing this restriction did not change the primary results of the model.) That restriction did not apply to the actor randomly chosen to be the actor holding the “focal idea” and randomly relocated in time period 1.
Figure OA2 displays the distribution of small world quotients across 100 initializations of each network in our study.
Figure OA2. Distribution of Small World Quotient Across Networks Used in Experiment
Notes: Y-axis is the small world quotient, defined as the ration
(C/CR)/( L/LR), where C is the clustering
coefficient for the observed network; CR the clustering
coefficient of a random network of the same size and connectivity; L,
the average path length for the observed network; and LR the
average path length of the corresponding random network.
- Change events. Actors interacting within the social network
produce five different types of change events as a consequence of
updating their current beliefs: (1) Self-search: actor learned about the
newly adopted idea through independent search of adjacent solutions; (2)
Within-network-community transfer: actor copied an idea from another
member of her own network community; (3) Between-network-community
transfer: actor copied an idea from a member of a different network
community; (4) Within-network-community recombination: actor developed
the idea through a recombination with another member of her own network
community; (5) Between-network-community recombination: actor developed
the idea through a recombination with a member of a different network
community.
Recombination, whether between or within a network community, occurs through single-point genetic crossover of the bit string representations of their beliefs (Holland and Order, 1995). The genetic crossover process yields a new location in feature space, whose performance each actor independently evaluates. Regardless of the type of attempted improvement, if the new location discovered is higher performing, an actor will update her belief. Single-point genetic crossover is accomplished by picking a random point between two digits, where each string gets “cut,” creating two pieces—left and right—for each actor. The left half of Actor A’s string is then concatenated with the right half of Actor B’s, and the right half of Actor A’s string is paired with the left half of Actor B’s to create two novel bit strings. We randomly selected one of the two bit strings produced for evaluation by the actors.
For example, consider the case where custom landscape is used; Actor A’s current belief for the highest performing location in feature space is {17, 25}, which corresponds to a bit string representation of 1000111001; Actor B’s current belief is location {28, 30}, which corresponds to a bit string representation of 1110011110. Further assume that the model chose a cut point between digits 6 and 7. Crossover in this case would yield the following two new strings 1000111110 and 1110011001. One of those two strings is then randomly selected (e.g., 1000111110), reconverted into the corresponding integer values in the feature space {28, 25}, and presented to both actors who independently compare the single outcome to their current belief. Recombination works in the exact same way when NK landscapes are used. The only difference is that the ideas are already represented as bit strings, so conversation from a spatial location to bit string is not necessary.
